First up, best dressed comes from big families; the first child awake wore the best clothes, so if you are first to do something, you are ahead or have an advantage. Similar to the early bird catches the worm.
(First in, best dressed is also used.)
When the gloves are off, people start to argue or fight in a more serious way.
('The gloves come off' and 'take the gloves off' are also used. It comes from boxing, where fighters normally wear gloves so that they don't do too much damage to each other.)